On the other hand, when granular materials are fully submerged, Cassar et al.[1]} proposed the viscous number, \(I_{v}=\eta _{f}\dot{\gamma }/P\) , where \(\eta _{f}\)  is the fluid dynamic viscosity. Boyer et al.[2]} generalized the \(\mu (I_{v})\)  constitutive law for suspensions. Trulsson et al.[3]} further investigated the rheology of submerged granular flows in the visco-inertial regime, where both contact forces and hydrodynamic forces play important roles, and proposed a combined dimensionless number \(K=\lambda I^{2}+I_{v}\)  for successfully describing the submerged granular flows in different flow regimes (by varying the viscosity of the interstitial fluid), but this work based on the assumption that the effect of fluid compared to the contact force is a constant represented by \(\lambda \)  in different conditions, and \(\lambda \)  is obtained by fitting. More effort is needed to establish a universal constitutive law suitable for the complex granular flow where both particle interactions and hydrodynamic forces are non-negligible.
In this case, a proper constitutive relationship is still lacking. In this work, we aim to tackle this problem using both theoretical derivations and numerical simulations.
